An Analysis of the Road Signs Classification Based on the Higher-Order Singular Value Decomposition of the Deformable Pattern Tensors - Advanced Concepts for Intelligent Vision Systems - page 193

where S k is a unitary mode matrix of dimensions N k × N k , spanning the column space

of the matrix T (k) obtained from the mode-n flattening of T ;

…… Z is a core tensor of the same dimensions as T , which satisfies
the following conditions [16]:
1. Two subtensors

NN N N N ×× × ×

∈ℜ

1 2 m n P

Z and Z , obtained by fixing the n k index to a , or b ,

k na

=

k nb

=

are orthogonal, i.e.

ZZ , ⋅ = 0

(2)

na nb = =

k k

for all possible values of k for which a ≠ b .

2. All subtensors can be ordered according to their Frobenius norms

ZZ Z ,

≥ ≥ … ≥ ≥ 0 (3)

n = 1 n = 2 n N =

k k k P

for all k .
The following Frobenius norm

k
na a

Z = σ (4)

=

k

is called the a-mode singular value of T . Each i-th vector of the matrix S k is the i-th k-
mode singular vector.
Assuming decomposition (1) of a tensor T , singular values (4) provide a notion of an
energy of this tensor in the terms of the Frobenius norm, as follows

R R

() ( )

2

1

2

P

2 2

∑ ∑

1 P

T = σ = = σ = Z , (5)

…

a a

a a

= 1 = 1

where R k denotes a k-mode rank of T .
The SVD decomposition allows representation of a matrix as a sum of rank one
matrices. The summation spans number of elements, however no more than a rank of
the decomposed matrix. Similarly to the SVD decomposition of matrices, based on
the decomposition (1), a tensor can be represented as the following sum

N

P

∑ s

T = T

h
hPP

× , (6)

h = 1

where

TZ =× × SS S × , (7)

…

h 1122 1 1 P − P −

denotes the basis tensors and s h P are columns of the unitary matrix S P . Since T h is of
dimension P -1 then

P in (6) is an outer product, i.e. a product of two tensors of
dimensions P -1 and 1 . The result is a tensor of dimension P , i.e. the same as of T .
Fig. 2 depicts a visualization of this decomposition for a 3D tensor. In this case T h

×